
 which is essentially the classical preconditioned iteration for solving the linear equation \(\mathbb {T}_k u=\mathfrak {b}^k\)  [1]}.
We thus reformulate the proximal iteration in () as the preconditioned iteration (REF ), which will turn out very useful.
 Through Lemma REF , we introduce the classical preconditioned iterations from numerical linear algebra and computation to the nonlinear alternating minimization by specially designed positive definite proximal terms \(M_k\)  and \(N_k\) . The idea can also be found in [2]}, [3]}. Here, for self-completeness, we give some explanation through a concrete example.
